An Algorithm for the Inversion of Finite Toeplitz Matrices

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Previous article Next article An Algorithm for the Inversion of Finite Toeplitz MatricesWilliam F. TrenchWilliam F. Trenchhttps://doi.org/10.1137/0112045PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] M. M. Siddiqui, On the inversion of the sample covariance matrix in a stationary autoregressive process, Ann. Math. Statist., 29 (1958), 585–588 MR0095568 0099.14305 CrossrefISIGoogle Scholar[2] J. Wise, The autocorrelation function and the spectral density function, Biometrika, 42 (1955), 151–159 MR0069468 0065.12902 CrossrefISIGoogle Scholar[3] Alberto Calderón, , Frank Spitzer and , Harold Widom, Inversion of Toeplitz matrices, Illinois J. Math., 3 (1959), 490–498 MR0121652 0091.11101 CrossrefGoogle Scholar[4] A. M. Yaglom, An introduction to the theory of stationary random functions, Revised English edition. Translated and edited by Richard A. Silverman, Prentice-Hall Inc., Englewood Cliffs, N.J., 1962xiii+235 MR0184289 0121.12601 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails High-Dimensional Gaussian Sampling: A Review and a Unifying Approach Based on a Stochastic Proximal Point AlgorithmMaxime Vono, Nicolas Dobigeon, and Pierre ChainaisSIAM Review, Vol. 64, No. 1 | 3 February 2022AbstractPDF (3135 KB)Hybrid Compact-WENO Finite Difference Scheme with Radial Basis Function Based Shock Detection Method for Hyperbolic Conservation LawsB. S. Wang, W. S. Don, Z. Gao, Y. H. Wang, and X. WenSIAM Journal on Scientific Computing, Vol. 40, No. 6 | 1 November 2018AbstractPDF (1188 KB)The Kähler Mean of Block-Toeplitz Matrices with Toeplitz Structured BlocksSIAM Journal on Matrix Analysis and Applications, Vol. 37, No. 3 | 1 September 2016AbstractPDF (370 KB)Approximate Inverse Circulant-plus-Diagonal Preconditioners for Toeplitz-plus-Diagonal MatricesSIAM Journal on Scientific Computing, Vol. 32, No. 3 | 21 May 2010AbstractPDF (542 KB)Interval Gaussian Elimination with Pivot TighteningSIAM Journal on Matrix Analysis and Applications, Vol. 30, No. 4 | 20 February 2009AbstractPDF (181 KB)Block Diagonal and Schur Complement Preconditioners for Block-Toeplitz Systems with Small Size BlocksSIAM Journal on Matrix Analysis and Applications, Vol. 29, No. 4 | 9 November 2007AbstractPDF (219 KB)Factorized Banded Inverse Preconditioners for Matrices with Toeplitz StructureSIAM Journal on Scientific Computing, Vol. 26, No. 6 | 25 July 2006AbstractPDF (795 KB)Recursive-Based PCG Methods for Toeplitz Systems with Nonnegative Generating FunctionsSIAM Journal on Scientific Computing, Vol. 24, No. 5 | 25 July 2006AbstractPDF (244 KB)The Best Circulant Preconditioners for Hermitian Toeplitz SystemsSIAM Journal on Numerical Analysis, Vol. 38, No. 3 | 26 July 2006AbstractPDF (233 KB)Multigrid Methods for Symmetric Positive Definite Block Toeplitz Matrices with Nonnegative Generating FunctionsSIAM Journal on Scientific Computing, Vol. 17, No. 5 | 17 February 2012AbstractPDF (1175 KB)Conjugate Gradient Methods for Toeplitz SystemsSIAM Review, Vol. 38, No. 3 | 12 July 2006AbstractPDF (6982 KB)Toeplitz-Circulant Preconditioners for Toeplitz Systems and their Applications to Queueing Networks with Batch ArrivalsSIAM Journal on Scientific Computing, Vol. 17, No. 3 | 17 February 2012AbstractPDF (1245 KB)Displacement Structure: Theory and ApplicationsSIAM Review, Vol. 37, No. 3 | 17 February 2012AbstractPDF (9852 KB)Moore–Penrose Inversion of Square Toeplitz MatricesSIAM Journal on Matrix Analysis and Applications, Vol. 15, No. 2 | 31 July 2006AbstractPDF (2505 KB)Generalized Displacement Structure for Block-Toeplitz, Toeplitz-Block, and Toeplitz-Derived MatricesSIAM Journal on Matrix Analysis and Applications, Vol. 15, No. 1 | 31 July 2006AbstractPDF (1450 KB)Fast Triangular Factorization and Inversion of Hermitian, Toeplitz, and Related Matrices with Arbitrary Rank ProfileSIAM Journal on Matrix Analysis and Applications, Vol. 14, No. 4 | 31 July 2006AbstractPDF (2276 KB)Fast Iterative Solvers for Toeplitz-Plus-Band SystemsSIAM Journal on Scientific Computing, Vol. 14, No. 5 | 13 July 2006AbstractPDF (751 KB)Circulant Preconditioners for Complex Toeplitz MatricesSIAM Journal on Numerical Analysis, Vol. 30, No. 4 | 14 July 2006AbstractPDF (1129 KB)The Use of Pivoting to Improve the Numerical Performance of Algorithms for Toeplitz MatricesSIAM Journal on Matrix Analysis and Applications, Vol. 14, No. 2 | 17 July 2006AbstractPDF (2506 KB)A Minimum-Phase LU Factorization Preconditioner for Toeplitz MatricesSIAM Journal on Scientific and Statistical Computing, Vol. 13, No. 6 | 13 July 2006AbstractPDF (1934 KB)Circulant Preconditioners Constructed from KernelsSIAM Journal on Numerical Analysis, Vol. 29, No. 4 | 14 July 2006AbstractPDF (973 KB)On Inversion of Symmetric Toeplitz MatricesSIAM Journal on Matrix Analysis and Applications, Vol. 13, No. 2 | 17 July 2006AbstractPDF (1098 KB)A Variant of the Gohberg–Semencul Formula Involving Circulant MatricesSIAM Jo